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Mirrors > Home > MPE Home > Th. List > fsnd | Structured version Visualization version GIF version |
Description: A singleton of an ordered pair is a function. (Contributed by AV, 17-Apr-2021.) |
Ref | Expression |
---|---|
fsnd.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
fsnd.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
Ref | Expression |
---|---|
fsnd | ⊢ (𝜑 → {〈𝐴, 𝐵〉}:{𝐴}⟶𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsnd.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | fsnd.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
3 | 1, 2 | jca 554 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) |
4 | f1sng 6178 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) | |
5 | f1f 6101 | . 2 ⊢ ({〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊 → {〈𝐴, 𝐵〉}:{𝐴}⟶𝑊) | |
6 | 3, 4, 5 | 3syl 18 | 1 ⊢ (𝜑 → {〈𝐴, 𝐵〉}:{𝐴}⟶𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 {csn 4177 〈cop 4183 ⟶wf 5884 –1-1→wf1 5885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 |
This theorem is referenced by: 1fv 12458 snopiswrd 13314 frgpcyg 19922 mat1dimmul 20282 pt1hmeo 21609 upgr1e 26008 1hevtxdg1 26402 wlkp1 26578 reprsuc 30693 breprexplema 30708 nnsum3primesprm 41678 |
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